Apr 24, 2023
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


On Effectivity in Some Diophantine Problems
Umberto Zannier (Scuola Normale Superiore)

 Location
 
 Video

 Abstract
I will survey on some issues concerning effectivity in diophantine problems. After a preamble of general nature, and a (very) brief summary of some of the known results, focusing mainly on integral points on curves, I shall present some new examples, especially concerning integral points on curves of genus 2. (Part of this concerns work in progress with P. Corvaja.)
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Integer Matrices with a Given Characteristic Polynomial and Multiplicative Dependence of Matrices
Alina Ostafe (University of New South Wales)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We consider the set $\mathcal{M}_n(\mathbb{Z}; H))$ of $n\times n$matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of $s$tuples of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb{Q})$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the noncommutativity of matrices. As a part of our method, we obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f \in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible.
Joint work with Igor Shparlinski.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Effective Counting and Families of Abelian Varieties
Harry Schmidt (Universität Basel)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In this talk, I will discuss the applications of counting techniques to unlikely intersection problems related to families of abelian varieties. After a very brief historical introduction, I will focus on questions related to effectivity and uniformity and, time permitting, extensions to fields of positive characteristics. In the course of my talk, I will cover joint work with Binyamini, Jones, and Thomas as well as, time permitting, Griffon.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Modularity and Effective Mordell
Levent Alpöge (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will give a finitetime algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many gdimensional abelian varieties A/K which are of GL_2type over K and have good reduction outside S.
The point of this is to effectively compute the Sintegral Kpoints on a Hilbert modular variety, and the point of that is to be able to compute all Krational points on complete curves inside such varieties.
This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height bounds for odddegree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group \Delta(3,6,6).
 Supplements




Apr 25, 2023
Tuesday

09:30 AM  10:30 AM


Bigness of the Admissible Canonical Bundle
Xinyi Yuan (Peking University)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
The admissible canonical bundle of a curve over a number field was first introduced by ShouWu Zhang and used to study the Bogomolov conjecture. In this talk, I will introduce a family version of the admissible canonical bundle, talk about its bigness, and deduce the uniform Bogomolov conjecture from the bigness. Previously, the uniform Bogomolov conjecture was proved by DimitrovGaoHabegger and Kuhne, and used together with Vojta's inequality to prove a uniform Mordell conjecture.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Preperiodic Points in Families of Rational Maps
Niki Myrto Mavraki (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Recently, Gao and Habegger have established the `relative ManinMumford conjecture' concerning the distribution of torsion points in families of abelian varieties. Inspired by the analogy between torsion points in abelian varieties and preperiodic points in a dynamical system, Zhang has proposed a dynamical analog of the ManinMumford conjecture. For instance, when can two rational maps share infinitely many common preperiodic points? In this talk we discuss progress towards answering a dynamical relative ManinMumford question, concerning the distribution of preperiodic points of families of rational maps. Though related questions have been considered in the dynamical setting by various authors, many problems remain unsolved. The talk will feature results with Harry Schmidt and with Laura DeMarco.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Reduction of Brauer Classes on K3 Surfaces
Salim Tayou (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Given a Brauer class on a K3 surface defined over a number field, I will prove that there exists infinitely many primes where the reduction of the Brauer class vanishes, under certain technical hypotheses. This answers a question of FreiHassettVárillyAlvarado. The proof relies on Arakelov intersection theory on integral models of GSpin Shimura varieties. The result of this talk is joint work with Davesh Maulik.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Computing the Weil Representation of a Superelliptic Curve
Irene Bouw (Ulm University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Let Y be a curve with potential good reduction to characteristic p>0. The local Galois representation of Y is determined by a Weil representation. We show how to determine this Weil representation from the stable reduction of Y. In the case of superelliptic curves, we make this explicit. This is joint work with D.K. Do and Stefan Wewers.
 Supplements



04:30 PM  06:20 PM


Reception

 Location
 SLMath: Front Courtyard
 Video


 Abstract
 
 Supplements




Apr 26, 2023
Wednesday

09:30 AM  10:30 AM


Heights in the Isogeny Class of an Abelian Variety
Mark Kisin (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous to A, and of height less than some fixed constant c.
In this talk, I will sketch a proof of the conjecture when the MumfordTate conjecture  which is known in many cases  holds for A. In particular, I will discuss finiteness in the case of a sequence of isogenies of pairwise coprime order.
This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.
While this talk will have some overlap with the one given at the introductory workshop it will *not* be the same.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Crystallinity Properties of Complex Rigid Local Systems (Joint work in progress with Michael Groechenig)
Hélène Esnault (Freie Universität Berlin)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We prove in all generality that on a smooth complex quasiprojective variety $X$, rigid connections yield $F$isocrystals on almost all good reductions $X_{\mathbb F_q}$ and that rigid local systems yield crystalline local systems on $X_K$ for $K$ the field of fractions of the Witt vectors of a finite field $\mathbb F_q$, for almost all $X_{\mathbb F_q}$. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$), and we obtained only infinitely many $\mathbb F_q$ of growing characteristic. While the earlier proof was via characteristic $p$, the new one is purely $p$adic and uses $p$adic topology.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Apr 27, 2023
Thursday

09:30 AM  10:30 AM


Degeneracy Loci in Families of Abelian Varieties and their Applications
Ziyang Gao (Leibniz Universität Hannover)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Given an abelian scheme over char 0 and an irreducible subvariety X, one can define the tth degeneracy locus of X for each integer t. This geometric concept of degeneracy loci has recently seen many applications in Diophantine Geometry, notably when t is 0 and 1, in the recent developments on the uniformity of the number of rational points on curves, on the solutions of the Uniform MordellLang Conjecture and of the Relative ManinMumford Conjecture. In this talk, I will define the degeneracy loci in the universal abelian variety, and explain how they are used in the applications mentioned above.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


A TwistingFree Converse Theorem for GL(2)
Vesselin Dimitrov (Institute for Advanced Study)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We explain how our joint work with Frank Calegari and Yunqing Tang further refines the unbounded denominators conjecture to take the form of a converse theorem for GL(2,A_Q) without any twists by Dirichlet characters. The thrust of such theorems is to replace Weil's character twists by a bounded denominators condition on the Dirichlet series coefficients. This opens up a view towards arithmetic algebraization methods for Dirichlet series, and motivates to search for integral converse statements for more general groups. We will conclude by discussing the situation for Hilbert modular groups.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Some New Elliptic Integrals
David Masser (Universität Basel)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
In 1981 James Davenport surmised that if an algebraic function $f(x,t)$ is not integrable (with respect to $x$) by elementary means when $t$ is an independent variable, then there are most finitely many complex numbers $\tau$ such that $f(x,\tau)$ is integrable by elementary means. In 2020 Umberto Zannier and I obtained a couple of counterexamples and in broad principle classified all of them with algebraic coefficients (they are necessarily somewhat rare). In this talk I will review our work, sketch our recent discovery of yet more counterexamples (they are not unrelated to Ribet curves), and give a more precise description of all elliptic counterexamples.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


A PAdic Analogue of Borel's Theorem
Ananth Shankar (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Shimura varieties are higher dimensional analogues of the modular curve. Borel proved that any holomorphic map from an affine complex algebraic curve to a Shimura variety (with sufficient level structure) must be algebraic. We will discuss a padic analogue of this theorem. The talk will mainly focus on the case of the moduli space of principally polarized abelian varieties. This is joint work with Abhishek Oswal and Xinwen Zhu, with an appendix by Anand Patel.
 Supplements




Apr 28, 2023
Friday

09:30 AM  10:30 AM


Conditional Algorithmic Mordell
Brian Lawrence (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Let X be a curve of genus at least 2 over a number field K. By Mordell's Conjecture, X(K) is known to be finite. I will discuss a (very slow) algorithmic approach to determining the set X(K), conditional on the Hodge, Tate, and FontaineMazur conjectures. Joint work in progress with Levent Alpöge.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Joint Unlikely Almost Intersections on Ordinary Siegel Spaces
Congling Qiu (Yale University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Motivated by the “Mordell–Lang + Bogomolov” (proved by Poonen and independently by S. Zhang), I hope to relax the usual incidence relations in unlikely intersections by using certain distance relations. The focus of this talk is a joint of the conjectures of Andr ́e–Oort and Andr ́e–Pink on a product of ordinary Siegel formal moduli schemes. I will make a conjecture and present partial progress obtained via a perfectoid approach. A closely related analog of the Ax–Lindemann principle will also be discussed.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Roth's Theorem Over Adelic Curves
Paul Vojta (University of California, Berkeley)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Recently, P. Dolce and F. Zucconi proved that Roth's theorem holds over adelic curves, under certain hypotheses.
Here, an adelic curve is a field $K$, together with a collection of absolute values parameterized by a measure space, such that an analogue of the product formula holds for all elements of $K^{*}$. (This definition was formulated by H. Chen and A. Moriwaki, and encompasses number fields, function fields, and Moriwaki's concept of arithmetic function fields.)
We will briefly describe adelic curves and some examples, and then summarize the proof of the theorem of Dolce and Zucconi.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Diophantine Geometry: All Our Yesterdays
ShouWu Zhang (Princeton University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In this talk, I will explain the impacts of some work of Szpiro, Edixhoven, and others who have recently left us.
 Supplements



